Question: Rewrite as a simplified fraction. ${1.89\overline{1} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1891.1111...\\ 100x &= 189.1111...\end{align*}}$ Subtracting the two equations, we have: ${900x = 1702}$ ${x = \dfrac{1702}{900}} $ ${x = \dfrac{851}{450}} \text{or} {1 \dfrac{401}{450}}$